#243918

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,10,4,7)(3,11,12,15,14,16,13,8)(17,18,21,26,23,27,22,37)(19,28,30,39,31,44,32,24)(20,25,35,42,36,43,34,41)(29,40,33,38,47,45,46,48);(2,7)(3,12)(4,5)(8,15)(9,10)(13,14)(17,19,22,30,23,31,21,32)(18,24,37,28,27,39,26,44)(20,33,34,47,36,46,35,29)(25,40,41,38,43,45,42,48);(1,3,6,14)(2,8,10,15)(4,12,5,13)(7,11,9,16)(17,20,23,36)(18,25,27,43)(19,29,31,47)(21,35,22,34)(24,38,39,48)(26,42,37,41)(28,45,44,40)(30,33,32,46);(1,4,6,5)(2,7,10,9)(3,13,14,12)(8,16,15,11)(17,21,23,22)(18,26,27,37)(19,30,31,32)(20,34,36,35)(24,28,39,44)(25,41,43,42)(29,46,47,33)(38,40,48,45)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151114);